91Ó°ÊÓ

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1 $$\int_{0}^{2}\left(x^{3}+x+1\right) d x$$

Variations on the substitution method Evaluate the following integrals. $$\int(z+1) \sqrt{3 z+2} d z$$

Derivatives of integrals Simplify the following expressions. $$\frac{d}{d t}\left(\int_{0}^{t} \frac{d x}{1+x^{2}}+\int_{1}^{1 / t} \frac{d x}{1+x^{2}}\right)$$

Derivatives of integrals Simplify the following expressions. $$\begin{array}{l} \frac{d}{d x} \int_{0}^{x} \sqrt{1+t^{2}} d t \\ \text { (Hint: }\left.\int_{-x}^{x} \sqrt{1+t^{2}} d t=\int_{-x}^{0} \sqrt{1+t^{2}} d t+\int_{-x}^{x} \sqrt{1+t^{2}} d t\right) \end{array}$$

Variations on the substitution method Evaluate the following integrals. $$\int x(x+10)^{9} d x$$

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1 $$\int_{0}^{1}\left(4 x^{3}+3 x^{2}\right) d x$$

Area by geometry Use geometry to evaluate the following integrals. $$\int_{1}^{6}|2 x-4| d x$$

Derivatives of integrals Simplify the following expressions. $$\frac{d}{d x} \int_{e^{x}}^{e^{2 x}} \ln t^{2} d t$$

Variations on the substitution method Evaluate the following integrals. $$\int_{0}^{\sqrt{3}} \frac{3 d x}{9+x^{2}}$$

Integrals with \(\sin ^{2} x\) and \(\cos ^{2} x\) Evaluate the following integrals. $$\int_{-\pi}^{\pi} \cos ^{2} x d x$$